mathsurgery - GCSE GM3

Modular GCSE Learning Outline created Oct 2010

module ref:	A4	topic:	Linear and curved graphs
week number:	year 10 wk 19-20	duration:	7 hours

Introduction:
This unit of work derives from Edexcel module 2-13 and Edexcel module 1-15. These originally were timetabled 5 - 9 hours and 5 - 7 hours respectively, so time will be tight, but pupils have covered much of this material in unit A2, so your first task will be to assess what they recall from that unit and work from there. Keep a careful note of the objectives you cover and plan to continue with this in mind next time. Within this fortnight you should also cover a lesson on real-life graphs, particularly using the Standards Unit tool on distance-time graphs and another lesson on modelling with graphs. This functional lesson should use skills of finding the equation of a straight line in the context of stretching springs in science or lines of best fit using the old coursework data set Mayfield High, perhaps.

Suggested Phasing of Lessons:

Lesson number	My chosen LOs	Starter	Main	Plenary
1	WILF is to find out what you can remember about linear graphs and linear sequences.	Baseline activity: find out what pupils can recall. Use MWB and hinge questions or the multiview activity?	Secure knowledge just within capability of average student.	Stretch, apply and extend.
2		Skill development: MWB or quizdom on the previous lesson’s (M)	Skill development: work toward secure knowledge of linear sequences, graphs and functions	(P) How might this be used in scatter graphs? Other subjects?
3	WALT use our knowledge and understanding of graphs to interpret distance-time (and velocity-time?) graphs	Suggested use of the materials in the Standards Unit box: traffic. There are teaching plans too.		sketch the speed time graph for a golf chip shot: video clip here
4		Skill development: check what is recalled from the previous lesson’s (M)	Skill development: work toward secure knowledge of next LO	(P) How might this be used in the real world? What next?
5		Skill development: check what is recalled from the previous lesson’s (M)	Skill development: work toward secure knowledge of next LO	(P) How might this be used in the real world? What next?
6		Skill development: check what is recalled from the previous lesson’s (M)	Skill development: work toward secure knowledge of next LO	(P) How might this be used in the real world? What next?
7	WALT use straight-line and non-linear graphs to model quantitive relationships in the real world. WILF is for you to find suitable functional relationship(s) to describe how the extension of a spring depends on the force applied to it.	Data collection or presentation		exceptions interpolation extrapolation

Modular GCSE Learning Outline created Oct 2010

module ref:	A4	topic:	Linear and curved graphs
week number:	year 10 wk 19-20	duration:	7 hours

grade	level	learning objective	learning activity
need sorting		¨*use the conventions for coordinates in the plane* ¨*plot points in all four quadrants* ¨*find the coordinates of the final point to make a given shape* ¨*use the formula to find the mid-point of line segments* ¨ ¨
A*		¨
A*		¨
A		¨
A		¨	ü http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=quadGraphs/sketchingQuadratics uses completing the square and transformation of graphs to make sophisticated predictions about the position and shape of quadratic graphs
B		¨
B		¨
C	7a	¨match simple quadratic, cubic and reciprocal graphs to their corresponding functions	Icard sort activity matching graphs to functions (to be designed and made) Êopportunity for paired or small group work ÊPLTs team worker – ‘work alongside others’ and ‘take part in discussions’ I10ticks level7/8 pk 3 pp39-42 – quadratic function snap cards ^ opportunity for paired or small group work ÊPLTs team worker – ‘work alongside others’ and ‘take part in discussions’ 8random graph generator with MWB feedback 8the Holt Online Graphing Calculator may be useful 8use graphical calculators to draw graphs to match desired screenshots ÊPLTs independent enquirer – ‘identify questions to answer and problems to sort out’ &Heinemann Higher pp369- 370 Ex 18D plot mixed functions 410ticks level7/8 pk 3 pp37-38 – plotting mixed functions H http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=graphs/recognisingGraphsOH ü http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=graphs/recognisingGraphs
	7b	¨plot graphs of simple quadratic and cubic functions, e.g. y = x2, y = 3x2 + 4, y = x3(APP L6 algebra item 5) ¨plot graphs of reciprocal functions such as y =	8plot a variety of graphs in Autograph to investigate the result of changing parameters in quadratic, cubic and reciprocal graphs: what changes the graph? ] opportunity for rich and open-ended task ÊPLTs independent enquirer – ‘analyse information and judge how important it is’ and ‘back up my conclusions using thoughtful arguments and reasons’ &Heinemann Higher pp369- 370 Ex 18A plot quadratics &Heinemann Higher pp369- 370 Ex 18B plot cubics &Heinemann Higher pp369- 370 Ex 18C plot reciprocals 410ticks level7/8 pk 3 pp29-30 – plotting quadratics 410ticks level7/8 pk 3 pp33-34 – plotting cubics 410ticks level7/8 pk 3 pp35-36 – plotting reciprocals ¸ Teachers’ TV video on quadratic graphs – shows the Kangaroo Kid using quadratics to model quad bike stunts @ opportunity to discuss other functional contexts in which quadratic, cubic and reciprocal functions might be used: @ Newtonian mechanics: equations of motion such as s = ut + ½at2 and universal law of gravitation g = Gm1m2/r2 @ link to ‘real-life graphs’ objective
	7c	¨pupils complete a table of values for coordinates using a non-linear rule
D		¨pupils use Cartesian coordinates for graphical representations, interpreting general features	ì History of Cartesian Coordinates and influence of Descartes ì
		¨pupils represent mappings expressed algebraically	8Standards Unit: Interpreting Graphs and Algebra Arrows ITP 8the Holt Online Graphing Calculator may be useful
		¨
E		¨
E		¨
F		¨
F		¨
G		¨
G		¨

D	¨link finding the intersection of the graphs of y = k and y = mx + c with solving the equation mx + c = k ¨consolidate graphs of linear functions, ensuring that there is basic understanding that c is the y‑intercept	NC6Use Cartesian coordinates for graphical representations, interpreting general features. from GCSE unit A2: NC6 ensure pupils can solve all sorts of linear equations including those with the variable on both sides and brackets. NC6 pupils order and approximate decimals when solving… equations using trial and improvement methods. [e.g. solve x³ + x = 20]


E	¨evaluate a range of formulae involving linear and then quadratic and more complicated polynomial and reciprocal terms - formulae should include suitable scientific formulae; substitute whole numbers first, then simple decimals – these should include written and calculator methods and might include a consideration of how to check answers by estimation ¨practise putting formulae given in words into algebra – perhaps use billing or wage calculations and volume or area formulae ¨Using and evaluating small positive whole number powers – find 53, write p × p × p × p as p4. ¨teach the order of operations using the mnemonic BODMAS, where ‘O’ stands for “Other stuff such as powers and roots” and care is taken to point out that DM and AS form pairs which should be evaluated simultaneously from left to right	NC5 construct, express in symbolic form and use simple formulae involving one or two operations.


F	¨define ‘reciprocal’ in respect of whole numbers in terms of ‘unit fractions’ ¨define ‘square numbers’ and ‘cube numbers’; ¨learn the first 10 square numbers, learn the first 5 cube numbers ¨explore the position to term formula tn = n2 and the term to term rule: “add two more each time” - relate these through pictures of squares ¨“continue the pattern 1, 4, 9, 16, 25 as far as you can in three minutes”; “how did you do that?”	NC4…recognise square numbers.


G	¨to access this unit at G grade, introduce the concept of using a letter to stand for a variable or a quantity; learn multiplication tables up to 10 ×10

GCSE Scheme of Work Higher skillsfor first teaching 2007-2009

Topic reference:

Topic title:

		PoS	NC level description
10a	A*		NCEPsolve problems using intersections… of graphs. NCEP simplify algebraic expressions using rules of indices for negative and fractional values.
10b
10c
9a	A
9b
9c
8a	B		NC8 sketch and interpret graphs of {linear,} quadratic, cubic and reciprocal functions. NC8 know that… a² + b² = (a + b)(a - b). NC8 manipulate algebraic formulae, equations and expressions, finding common factors and multiplying {any} two linear expressions. NC8 solve problems involving calculating with powers {and} roots. NC8 calculate one variable given the others in formulae such as V = π r² h. NC8 evaluate algebraic formulae, substituting fractions, decimals and negative numbers.
8b
8c		¨evaluate increasingly complicated (scientific and mathematical) formulae, with a wide range of (mostly rational) variables, rounding at the end to an appropriate number of decimal places; ¨this might be a good opportunity to go on to review calculations in standard index form ¨simple linear factorisation to give a(x±b) ¨harder linear factorisation with powers, for example 3x3 + 2x2y = x2(3x + 2y)
7a	C	¨expand pairs of brackets of the form(x ±a)(x±b) ¨pay special attention to the case with two minus signs ¨PCa calls this the ‘Gallagher method’ after the Oasis brothers with big eyebrows; alternatively use box method ¨check that pupils can reliably expand a(bx ± c) ¨PCa calls this the Elvis’ Quiff method ¨match simple quadratic, cubic and reciprocal graphs to their corresponding functions: card sort activity? ¨plot graphs of simple quadratic, cubic and reciprocal functions from values in tables ¨pupils might start by completing a table that is partially filled-in	NC7multiply two expressions of the form (x + n) and simplify the corresponding quadratic expression.
7b
7c
6a	D	¨link finding the intersection of the graphs of y = k and y = mx + c with solving the equation mx + c = k ¨consolidate graphs of linear functions, ensuring that there is basic understanding that c is the y‑intercept revise solving a wide variety of linear equations basic rules of indices for positive integer powers explain why it is necessary to continue to consider a ‘5’ in the (n+1)th decimal position in order to solve an equation to n decimal places by trial and improvement ; show this situation on a number line trial and improvement to solve equations involving quadratic, cubic or reciprocal terms - set out in a table showing, in the left hand column, each guess for x; columns of working and the value of f(x); and, in the right hand column, “too big” or “too small”	NC6represent mappings expressed algebraically. Use Cartesian coordinates for graphical representations, interpreting general features. from GCSE unit A2: NC6 ensure pupils can solve all sorts of linear equations including those with the variable on both sides and brackets. NC6 pupils order and approximate decimals when solving… equations using trial and improvement methods. [e.g. solve x³ + x = 20]
6b
6c

Modular GCSE Learning Outline created Oct 2010

module ref:	A4	topic:	Linear and curved graphs
week number:	year 10 wk 19-20	duration:	7 hours

Lesson number	My chosen LOs	Starter	Main	Plenary
1	WILF is to find out what you can remember about linear graphs and linear sequences.	Baseline activity: find out what pupils can recall. Use MWB and hinge questions or the multiview activity?	Secure knowledge just within capability of average student.	Stretch, apply and extend.
2		Skill development: MWB or quizdom on the previous lesson’s (M)	Skill development: work toward secure knowledge of linear sequences, graphs and functions	(P) How might this be used in scatter graphs? Other subjects?
3	WALT use our knowledge and understanding of graphs to interpret distance-time (and velocity-time?) graphs	Suggested use of the materials in the Standards Unit box: traffic. There are teaching plans too.		sketch the speed time graph for a golf chip shot: video clip here
4		Skill development: check what is recalled from the previous lesson’s (M)	Skill development: work toward secure knowledge of next LO	(P) How might this be used in the real world? What next?
5		Skill development: check what is recalled from the previous lesson’s (M)	Skill development: work toward secure knowledge of next LO	(P) How might this be used in the real world? What next?
6		Skill development: check what is recalled from the previous lesson’s (M)	Skill development: work toward secure knowledge of next LO	(P) How might this be used in the real world? What next?
7	WALT use straight-line and non-linear graphs to model quantitive relationships in the real world. WILF is for you to find suitable functional relationship(s) to describe how the extension of a spring depends on the force applied to it.	Data collection or presentation		exceptions interpolation extrapolation

Modular GCSE Learning Outline created Oct 2010

module ref:	A4	topic:	Linear and curved graphs
week number:	year 10 wk 19-20	duration:	7 hours

grade	level	learning objective	learning activity
need sorting		¨*use the conventions for coordinates in the plane* ¨*plot points in all four quadrants* ¨*find the coordinates of the final point to make a given shape* ¨*use the formula to find the mid-point of line segments* ¨ ¨
A*		¨
A*		¨
A		¨
A		¨	ü http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=quadGraphs/sketchingQuadratics uses completing the square and transformation of graphs to make sophisticated predictions about the position and shape of quadratic graphs
B		¨
B		¨
C	7a	¨match simple quadratic, cubic and reciprocal graphs to their corresponding functions	Icard sort activity matching graphs to functions (to be designed and made) Êopportunity for paired or small group work ÊPLTs team worker – ‘work alongside others’ and ‘take part in discussions’ I10ticks level7/8 pk 3 pp39-42 – quadratic function snap cards ^ opportunity for paired or small group work ÊPLTs team worker – ‘work alongside others’ and ‘take part in discussions’ 8random graph generator with MWB feedback 8the Holt Online Graphing Calculator may be useful 8use graphical calculators to draw graphs to match desired screenshots ÊPLTs independent enquirer – ‘identify questions to answer and problems to sort out’ &Heinemann Higher pp369- 370 Ex 18D plot mixed functions 410ticks level7/8 pk 3 pp37-38 – plotting mixed functions H http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=graphs/recognisingGraphsOH ü http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=graphs/recognisingGraphs
	7b	¨plot graphs of simple quadratic and cubic functions, e.g. y = x2, y = 3x2 + 4, y = x3(APP L6 algebra item 5) ¨plot graphs of reciprocal functions such as y =	8plot a variety of graphs in Autograph to investigate the result of changing parameters in quadratic, cubic and reciprocal graphs: what changes the graph? ] opportunity for rich and open-ended task ÊPLTs independent enquirer – ‘analyse information and judge how important it is’ and ‘back up my conclusions using thoughtful arguments and reasons’ &Heinemann Higher pp369- 370 Ex 18A plot quadratics &Heinemann Higher pp369- 370 Ex 18B plot cubics &Heinemann Higher pp369- 370 Ex 18C plot reciprocals 410ticks level7/8 pk 3 pp29-30 – plotting quadratics 410ticks level7/8 pk 3 pp33-34 – plotting cubics 410ticks level7/8 pk 3 pp35-36 – plotting reciprocals ¸ Teachers’ TV video on quadratic graphs – shows the Kangaroo Kid using quadratics to model quad bike stunts @ opportunity to discuss other functional contexts in which quadratic, cubic and reciprocal functions might be used: @ Newtonian mechanics: equations of motion such as s = ut + ½at2 and universal law of gravitation g = Gm1m2/r2 @ link to ‘real-life graphs’ objective
	7c	¨pupils complete a table of values for coordinates using a non-linear rule
D		¨pupils use Cartesian coordinates for graphical representations, interpreting general features	ì History of Cartesian Coordinates and influence of Descartes ì
		¨pupils represent mappings expressed algebraically	8Standards Unit: Interpreting Graphs and Algebra Arrows ITP 8the Holt Online Graphing Calculator may be useful
		¨
E		¨
E		¨
F		¨
F		¨
G		¨
G		¨

D	¨link finding the intersection of the graphs of y = k and y = mx + c with solving the equation mx + c = k ¨consolidate graphs of linear functions, ensuring that there is basic understanding that c is the y‑intercept	NC6Use Cartesian coordinates for graphical representations, interpreting general features. from GCSE unit A2: NC6 ensure pupils can solve all sorts of linear equations including those with the variable on both sides and brackets. NC6 pupils order and approximate decimals when solving… equations using trial and improvement methods. [e.g. solve x³ + x = 20]


E	¨evaluate a range of formulae involving linear and then quadratic and more complicated polynomial and reciprocal terms - formulae should include suitable scientific formulae; substitute whole numbers first, then simple decimals – these should include written and calculator methods and might include a consideration of how to check answers by estimation ¨practise putting formulae given in words into algebra – perhaps use billing or wage calculations and volume or area formulae ¨Using and evaluating small positive whole number powers – find 53, write p × p × p × p as p4. ¨teach the order of operations using the mnemonic BODMAS, where ‘O’ stands for “Other stuff such as powers and roots” and care is taken to point out that DM and AS form pairs which should be evaluated simultaneously from left to right	NC5 construct, express in symbolic form and use simple formulae involving one or two operations.


F	¨define ‘reciprocal’ in respect of whole numbers in terms of ‘unit fractions’ ¨define ‘square numbers’ and ‘cube numbers’; ¨learn the first 10 square numbers, learn the first 5 cube numbers ¨explore the position to term formula tn = n2 and the term to term rule: “add two more each time” - relate these through pictures of squares ¨“continue the pattern 1, 4, 9, 16, 25 as far as you can in three minutes”; “how did you do that?”	NC4…recognise square numbers.


G	¨to access this unit at G grade, introduce the concept of using a letter to stand for a variable or a quantity; learn multiplication tables up to 10 ×10

GCSE Scheme of Work Higher skillsfor first teaching 2007-2009

Topic reference:

Topic title:

		PoS	NC level description
10a	A*		NCEPsolve problems using intersections… of graphs. NCEP simplify algebraic expressions using rules of indices for negative and fractional values.
10b
10c
9a	A
9b
9c
8a	B		NC8 sketch and interpret graphs of {linear,} quadratic, cubic and reciprocal functions. NC8 know that… a² + b² = (a + b)(a - b). NC8 manipulate algebraic formulae, equations and expressions, finding common factors and multiplying {any} two linear expressions. NC8 solve problems involving calculating with powers {and} roots. NC8 calculate one variable given the others in formulae such as V = π r² h. NC8 evaluate algebraic formulae, substituting fractions, decimals and negative numbers.
8b
8c		¨evaluate increasingly complicated (scientific and mathematical) formulae, with a wide range of (mostly rational) variables, rounding at the end to an appropriate number of decimal places; ¨this might be a good opportunity to go on to review calculations in standard index form ¨simple linear factorisation to give a(x±b) ¨harder linear factorisation with powers, for example 3x3 + 2x2y = x2(3x + 2y)
7a	C	¨expand pairs of brackets of the form(x ±a)(x±b) ¨pay special attention to the case with two minus signs ¨PCa calls this the ‘Gallagher method’ after the Oasis brothers with big eyebrows; alternatively use box method ¨check that pupils can reliably expand a(bx ± c) ¨PCa calls this the Elvis’ Quiff method ¨match simple quadratic, cubic and reciprocal graphs to their corresponding functions: card sort activity? ¨plot graphs of simple quadratic, cubic and reciprocal functions from values in tables ¨pupils might start by completing a table that is partially filled-in	NC7multiply two expressions of the form (x + n) and simplify the corresponding quadratic expression.
7b
7c
6a	D	¨link finding the intersection of the graphs of y = k and y = mx + c with solving the equation mx + c = k ¨consolidate graphs of linear functions, ensuring that there is basic understanding that c is the y‑intercept revise solving a wide variety of linear equations basic rules of indices for positive integer powers explain why it is necessary to continue to consider a ‘5’ in the (n+1)th decimal position in order to solve an equation to n decimal places by trial and improvement ; show this situation on a number line trial and improvement to solve equations involving quadratic, cubic or reciprocal terms - set out in a table showing, in the left hand column, each guess for x; columns of working and the value of f(x); and, in the right hand column, “too big” or “too small”	NC6represent mappings expressed algebraically. Use Cartesian coordinates for graphical representations, interpreting general features. from GCSE unit A2: NC6 ensure pupils can solve all sorts of linear equations including those with the variable on both sides and brackets. NC6 pupils order and approximate decimals when solving… equations using trial and improvement methods. [e.g. solve x³ + x = 20]
6b
6c

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GCSE GM3